1 A STRATEGIC APPROACH TO ASSESSING NEEDS AND ENGINEERING COST/PERFORMANCE TRADEOFFS FOR INTERNALLY DEVELOPED HIGH PERFORMANCE ASSEMBLY MACHINES by Jefferson Davis Hamlin, Jr. B.S., Mechanical Engineering, University of Virginia (1986) Submitted to the Department of Mechanical Engineering and to the MIT Sloan School of Management in Partial Fulfillment of the Requirements for the Degrees of MASTER OF SCIENCE IN MECHANICAL ENGINEERING and MASTER OF SCIENCE IN MANAGEMENT at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 1994 Copyright Massachusetts Institute of Technology, 1994 All rights reserved -- i/ 11 // Signature of Author I Lbepartment ofk echanical Engineering MIT Sloan School of Management May 6, 1994 I J A. I, Certified by Certified by Accepted by Haruhiko Asada, Professor of Mechanical Engineering Thesis Supervisor Lawrence Wein, Assqqih' i Professor of Management Science Thesis Supervisor,%"ra, - Jeffrey Barks Associate I an "h aster' and Bachelor's Prograns MASSACHUSETTS. INSTITUTE OF * NWOtOGY rjun LIBRARIES
3 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for Internally Developed High Performance Assembly Machines by Jefferson Davis Hamlin, Jr. Submitted to the Department of Mechanical Engineering and to the MIT Sloan School of Management in Partial Fulfillment of the Requirements for the Degrees of Master of Science in Mechanical Engineering and Master of Science in Management Abstract A case study of two internally developed assembly machines is presented. Based upon this work, a decision framework is proffered for determining strategic ramifications and engineering performance requirements for assembly machines. This is accomplished by examining a firm's core competencies and dividing its manufacturing process into individual tasks. The framework also addresses the make versus buy decision in this context. In the event that a firm elects to develp its own manufacturing equipment, a parametric procedure is described for ascertaining the effect of various components on machine performance. This technique requires the determination of the machine's transfer function which can either be modeled using lumped parameters or experimentally determined. Both procedures are described. Thesis Advisors: Dr. Haruhiko Asada, Professor of Mechanical Engineering Dr. Lawrence Wein, Associate Professor of Management Science
5 The author wishes to acknowledge the Leaders for Manufacturing Program. The author also wishes to acknowledge the guidance and counsel of his advisors Harry Asada and Larry Wein. Without the help and assistance of Sunil Lakhani, Rick Marquardt, Stan Renteria, and Dan Yeaple this thesis would not have been possible.
7 Resource Leverage. Table of Contents Chapter 1 - Introduction... 8 Acme Mfg.'s Competitive Situation... 8 Literature Survey Chapter 2 - The Acme Mfg. Gantry Robot... Gantry Description Chapter 3 - Development of a Solution Methodology... Problem Statement Solution M ethodology C hoice of G oals... Settling Time... Complications Introduced by Closed Loop Feedback Systems... The Root-Locus Plot... Calculating Kp and Avoiding Saturation... Calculating Unit Manufacturing Cost... Chapter 4 - Application of the Methodology... M odeling the Plant... Identification of the Plant M odeling the Controller Using the Root Locus to Determine KI, K, and the Settling Time... Bode Plots... Block LLS Identifier Final Results... Chapter 5 - The Second Generation... Lessons Learned... Chip Placement Machines... The Second Generation A Rigid Structure Chapter 6 - Manufacturing Strategy - Assessing Automation Needs... Costs and Benefits... The Product Development Process Manufacturing and Engineering Core Competencies oo.ooooo. o..o
8 Make versus Buy Transfer of Core Competencies M odularization M aking the Decision Benefits of the Three Dimensional Array Chapter 7 - Applying the Three Dimensional Array Adding a Product to the Three Dimensional Array Competency Resolution Final Analysis Making the Make versus Buy Decision Chapter 8 - Conclusion Design Considerations for High Performance Assembly Machines On Costs and Strategic Considerations... 84
9 Chapter I - Introduction Acme Mfg.'s Competitive Situation The research for this thesis was conducted at a large corporation that wishes to remain anonymous. The company has been disguised as Acme Mfg., a fictitious entity. Acme produces widgets, a high-tech electronics product that requires substantial use of surface mount technology (SMT). The trend towards ever increasing global competition has forced many manufacturing companies to rethink the way they approach manufacturing. Acme Mfg. Widgets Products Group, WPG, long the global leader in the design and manufacture of widgets recently faced the possibility of being cut-off from critical manufacturing technology. Their current manufacturing process relies heavily upon chip placement machines supplied by external vendors. The entry of one these suppliers into the widgets market caused great consternation among Acme Mfg.'s top management and a program was initiated to reduce Acme Mfg.'s reliance upon externally supplied proprietary manufacturing technology. As a result of this initiative, Acme Mfg. has embraced an open architecture for its factories and has tried to standardize upon a single technology in any given category. For example, Acme Mfg. has chosen UNIX as the preferred technology in the operating system category. Similarly, they have elected to program in C++ and communicate via Ethernet. Because all of these products are supplied by a large number of vendors, Acme Mfg. believes that competition among vendors will insure technological advancement and
10 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for price reduction. It is believed that standardization will reduce complexity and training costs. In the fields of robotics and chip placement machines, Acme Mfg. could not find a vendor willing to supply a non-proprietary system. Thus standardization meant being at the mercy of a single supplier. The Widgets Products Group felt that they had already done their share of debugging vendors products and wanted to "control their own destiny in the future."' Additionally, WPG realized there were a number of material handling tasks that did not require the accuracy provided by the vendor's robots. Unable to find a market mechanism to insure price competitiveness and product effectiveness, Acme Mfg. decided to study the feasibility of building their own robots. The feasibility study included a survey of alternatives to the vendor's robots and a survey of vendors who produced components used in the construction of robots. These surveys confirmed two ideas that Acme Mfg. held. First, there was not an acceptable alternative to the vendor's robots, and second, the cost of the sum of the components was substantially less than the cost of a completed robot. Typically, robot manufacturers only produce the structure and controller while the motors, encoders, bearings, amplifiers, and power supplies are purchased from external suppliers. Acme Mfg.'s Automation Engineering group had been extremely successful in integrating technologies to produce automated factories. Furthermore, the engineers were very adept at writing the computer coding necessary to create an automated factory. Acme Mfg. felt the construction of a robot entailed the integration of components, the Interview by the author with an Acme Mfg employee. 10
11 Internally Developed High Performance Assembly Machines Hamlin generation of software, and the creation of a structure. In this light, the construction of a robot seemed well within their capabilities. Although Acme Mfg. was essentially correct in this estimation, they vastly under-estimated the skills required to build a high performance machine from the components. Viewing systems integration as their strength, Acme Mfg. elected to purchase the controller and to build the structure out of aluminum extrusions. The vendor of aluminum extrusions also produces a full line of bolted connectors and machine components for the construction of manufacturing equipment and work stations. The chief virtue of this product line is its "erector set" ease of producing machinery. Because of Acme Mfg.'s skill in integration, software development, and project management, they were able to produce a working gantry robot in the remarkably short period of 6 months. However, the machine did not fulfill all of Acme Mfg.'s expectations and Acme Mfg. decided to offer the improvement of the gantry robot to MIT as a master's thesis project. As a fellow in the Leader's for Manufacturing Program, I was assigned the task of improving the performance of this gantry robot. The initial inspection of the machine found the structure to be too compliant. Acme Mfg. had attempted to reduce this compliance by adding corner braces where possible. Although these braces proved somewhat effective, they did not achieve the performance goals that Acme Mfg. desired. Redesigning the structure to improve rigidity was recommended, but Acme Mfg. wanted to retain the ability to size the machine for different tasks. If possible, Acme Mfg.
12 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for wanted to develop a control system to compensate for the compliance in the structure, thus enabling them to retain a scaleable structure. This thesis introduces a methodology for improving the machine's performance by comparing the incremental changes in performance, caused by changing a single component of the system. This methodology is introduced in chapter 3 and is applied to the Acme Mfg. gantry in chapter 4. Chapter 2 describes the gantry robot in detail. Although the methodology proved to be moderately successful at improving the performance of the gantry robot, the performance was ultimately limited by a number of factors. Based upon these factors, a second robot was designed considering both the structure and the control system. This machine is described in chapter 5. Finally, a methodology to assist management in the assessment of their strategic manufacturing needs is proffered in Chapter 6. This methodology provides management a new perspective on the strategic interrelationships between manufacturing and design competencies. The perspective offered can be used to assess manufacturing needs, conduct competitive analyses, find areas of opportunity, uncover weaknesses, and aid in the make versus buy decision. An example of these usages is provided in chapter 7. Literature Survey The machine improvement strategy introduced in this paper is based upon prior work in the field of space-structure design. A subset of this field is integrated structure/controller design. The original work in integrated structure/controller design was conducted in order to develop robotic manipulators for the space program, and is characterized by various mathematical optimization techniques. Typically, a model is 12
13 Internally Developed High Performance Assembly Machines Hamlin used to describe the dynamic behavior of the system, and an analysis is conducted which aims to maximize a performance function while minimizing a cost function. The subject of space-structure design is extensively covered in Eschnauer et. al.[l]. Integrated structure-controller design has been applied to the problem of minimizing the weight of a machine structure subject to modal damping equality constraints and cross-sectional area inequality constraints of the structural members by Khot, Venkayya et. al.. Miller and Shim used a gradient base technique to optimize the sum of structural and control parameters. The problem of obtaining a set of control forces and structural parameters that minimize a specific cost function has been solved by Hale, Lisowski et. al.. An eigenspace optimization technique which simultaneously optimizes over a machine's structural and control parameters has been developed by Bodden and Junkins. A computer-aided procedure has been developed by Rai and Asada that decomposes the sensitivity matrix in order to aid the designer in the selection of an optimal geometry, and has been applied to a flexible arm. A concurrent structure/control design method has been used by Park and Asada to optimize the performance of a high speed two link robot. The methodology proposed herein differs from prior work by focusing on the unit manufacturing costs of the product produced by the machine rather than the cost functions of the machine, and by evaluating the performance of purchased components rather than design parameters.
14 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for The strategic assessment methodology proposed is based upon work by Hamel and Prahalad in core competencies of a firm and in resource leverage. A framework for applying these concepts simultaneously to a firm's product offering is described. The framework consists of constructing a three dimensional array based upon a firm's products, design needs, and manufacturing tasks.
15 -Internally Developed High Performance Assembly Machines Hamlin Chapter 2 - The Acme Mfg. Gantry Robot Gantry Description The Acme Mfg. gantry robot is shown in figure 2.1. As mentioned above, the primary design goal of this machine is to provide Acme Mfg. with a means of quickly constructing automated manufacturing cells. A key feature of the gantry design is its scaleability. This is accomplished through the use of aluminum extrusions and bolted joints. Unfortunately, this asset is also one of the machine's greatest liabilities because the construction technique fails to provide the required rigidity. As of May 1994, twenty of these machines had been built in five different sizes for conducting five different assembly tasks. These machines share a number of common features. They are capable of generating four degrees of freedom: translations in the x, y, and z directions and rotation about the z axis, which is defined as the theta (0) axis. Starting at the end effector and working back to the structure, the gantry robot consists of a number of purchased components. The end effector is held by a tool changer that allows the machine to use multiple end effectors. The tool changer, in turn, is connected to a device called a break-away-joint. A break-away-joint shuts down the machine when a collision is detected between the end effector and an unexpected obstacle. This device provides insurance for some very expensive end effector tooling. Unfortunately, the device also adds weight to every axis of motion. The break-away-joint is connected to a 50:1 harmonic drive that is driven by a brushless DC motor. This motor/transmission pair provides motion about the 0 axis and
16 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for Pulle Conrlectin Figure Acme Mfg. Gantry Robot
17 Internally Developed High Performance Assembly Machines Hamlin is used to actuate rotational changes of the end effector. The harmonic drive is mounted at the end of the z-axis aluminum extrusion and the motor and its resolver reside inside of the extrusion. The z-axis is mounted to the y-axis using a recirculating ball linear bearing. The ball bearing slider cars are mounted on the y-axis carriage plate and the ground rail is bolted to the z-axis aluminum extrusion. A pneumatic counter balance is used to support the weight of the z-axis. A brushed permanent magnet DC servo-motor turning a 10:1 gearbox is used to raise and lower the z-axis via a toothed belt and pulley arrangement. Position sensing is handled by a rotary optical encoder mounted on the motor shaft. The y-axis carriage plate rides on a pair of recirculating ball linear bearings mounted on the top and bottom of the y-axis extrusion. A looped belt is connected to two sides of the carriage plate. The looping is accomplished using an idler pulley and a drive pulley. The drive pulley is driven by a 5:1 gearbox and a brushed permanent magnet DC servo-motor. Position sensing is accomplished by a rotary encoder mounted on the motor shaft and/or a linear encoder mounted along the y-axis aluminum extrusion. The y-axis is a bridge arrangement and is therefore supported on both ends. Each end is bolted to a carriage plate that rides on a recirculating ball linear bearing. These bearings make up the x-axis which also consists of two aluminum extrusions and a pair of looped belts to drive the two carriage plates. A brushed permanent magnet motor driving a 5:1 planetary gear reducer drives one of the drive pulleys which turns a long drive shaft that drives the other drive pulley. The two idler pulleys are not connected. The desired effect of this
18 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for arrangement is to maintain orthogonality between the x and y axes during translation of the y-axis. The x-axis is mounted approximately six feet above the ground, supported by four aluminum extrusions. The four aluminum columns are braced by four horizontal members and a variety of angle braces depending upon the installation requirements. This structure also supports the electrical cabinetry that houses the servo-amplifiers, power supply, transformer, and safety system. The control scheme consists of a VME bus-based computer that acts as a cell controller, communicating with the factory host via an ethernet card. The cell controller can control two gantry robots and their end effector tooling. The actual servo control is handled by a pair of 4-axis vendor supplied controller cards, one for each gantry. These controllers are only capable of producing trapezoidal trajectories. The control signals are amplified by pulse width modulation amplifiers that operate in torque mode.
19 Internally Developed High Performance Assembly Machines Hamlin Problem Statement Chapter 3 - Development of a Solution Methodoloav Although the initial machines did not meet Acme Mfg.'s design goals for speed and accuracy, the machines successfully performed their intended task. Because the operation took longer than expected, a reduction in cycle time was required to keep pace with the rest of the automated factory. Similarly, the lower than desired accuracy of the machine precluded its use in a number of Acme Mfg.'s manufacturing operations. A solution that required minimal re-design of the existing structure was desired, but the limited programmability of the original controller made a controls solution all but impossible. As a work around, Acme Mfg. elected to experiment with another controller which provided increased programmability. Solution Methodology First it must be stated that in order to achieve high performance from an assembly machine, the design must successfully integrate both the structure and control design. The old approach of producing a static and kinematic design of the structure followed by a design of the control system fails to account for the intricate interactions that occur between the structure and the control system. 2 As the speed and accuracy requirements of the machine are raised, these effects becoming increasingly important. Thus, a solution methodology that only addresses the control sub-system of the assembly machine is 2 Jahng-Hyon Park and Haruhiko Asada. "Concurrent Design Optimization of Mechanical Structure and Control for High Speed Robots." (Accepted for publication in the ASME Journal of Dynamic Systems, Measurement, and Control).
20 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for limited in its ability to correct the problem. It should be noted that for any linear third order or higher system, there exists an optimal control solution. The solution strategy was to model the existing machine using linear elements and then find the optimal control solution. Because the Acme Mfg. engineers had most likely already found or were at least near the optimal controller proportional and velocity gains (Kp and K,), there was little reason to believe that a mathematical determination of the optimal gains would provide a significant performance improvement. On the other hand, once the mathematical model had been constructed the performance effects of possible component substitutions could be predicted. For example, a stiffer transmission could be tested by changing the model to reflect the new transmission's stiffness and damping characteristics. The basic idea was to find the most cost effective component substitutions that could be made to improve the performance. This technique differs from a typical parametric design study which would determine the effect of each dynamic parameter upon the desired performance goal. It was decided to perform the analysis for components because many components affected multiple dynamic parameters and because it was much easier to compute the costs. The solution methodology therefore consists of the following steps: 1. Model the plant 2. Model the controller 3. Find the optimal values of K, and Kv, the values that produce the fastest settling, by using the root locus plot 4. Change a component of the machine 5. Find optimal values of K, and K, for the modified machine 6. Repeat steps 4 & 5 for all components under consideration
21 Internally Developed High Performance Assembly Machines 7. Based upon the real part of the root found in steps 3 or 5, calculate the cycle time for completing the manufacturing operation 8. Based upon the cost of the component, determine which component provides the lowest unit manufacturing cost where the unit manufacturing cost is the annualized cost of the machine divided by annual total production. If the annual total production is unknown, the theoretical capacity of the machine may be substituted. If none of the components improve the unit manufacturing cost, then the current basecase is the optimal set of components. 9. Add this component to the basecase, eliminate all other choices for this component, and repeat steps 3 through 9 for the remaining component choices. Continue until an iteration does not improve the unit manufacturing cost. The basecase of this iteration is the optimal set of components. Hamlin Choice of Goals The design goal for this methodology is to minimize the settling time as cost effectively as possible. The selection of this goal stems from the general underlying goal of all assembly machines, which is to bring two items together and then effect some change in their relationship to each other as cheaply as possible. In other words, the general goal is to effect a relative change in the position and relationship of two items as cheaply as possible. The two items could be a chip and a printed circuit board, the two halves of a housing, or a windshield and an adhesive dispensing nozzle. It is instructive to think of the assembly process as consisting of a positioning step and an operation-at-point step. The positioning step is used to bring about a relative change in position of the two items, and the operation-at-point step effects a change in the relationship between the two items. For example, in the case of the printed circuit board and chip, the position step is the motion required to move the chip from where it was
22 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for picked up to its correct location and orientation on the printed circuit. The operation-at point-step is the physical tacking of the chip to the board via the solder paste. The relationship change is from not being attached to being attached. The machine's cycle time is the sum of the times required to complete all of the positioning steps and operation-at-point steps. It is assumed that the operation-at-point step's cycle time is fixed and not affected by the structure or control. The time a machine takes to complete an operation is inversely proportional to the number of operations a machine can complete in a given amount of time. Thus, the manufacturing costs on a per unit basis decrease as the cycle time decreases. Since this relationship always hold true for a given set of components and there is no cost associated with changing the controller gains, the most efficient operating point for any set of components is the operating point that minimizes cycle time, assuming no changes in machine reliability. For these reasons, the design goal was to minimize settling time as cost effectively as possible. Settling Time Most high precision assembly processes consist of a stationary item and a dynamic item. In order to perform the operation, the dynamic item must be accelerated from rest, translated and rotated, and brought to rest in the desired position and orientation. The objective is not raw speed but rather a short cycle time to complete the operation. In the language of dynamic systems, this goal is referred to as minimizing the settling time. Settling time is defined as "the time required for the response curve to 22
23 -Internally Developed High Performance Assembly Machines Hamlin reach and stay within a range about the final value of size specified by absolute percentage of the final value (usually 2% or 5%)." ' 3 A graphical representation is given in figure 3.1. Definition of Settling Time 1A / V Arbitrary Percentage (frequently chosen to be 2% or 5%) The settling time is defined as how long it takes for the waveform to achieve a condition whereby it remains within an arbitrary specified percentage of the desired position Settling Time Time (seconds) Figure,, l - Definition of Settling Time Complications Introduced by Closed Loop Feedback Systems The choice of a closed loop feedback control system adds to the complexity of predicting the settling time because the system can oscillate about the desired final position as shown in figure 3.1. An introduction to closed loop feedback systems is given in Appendix 1. It should be noted that these oscillations are a function of the dynamic parameters of the machine and the gains of the controller. The closed loop feedback system functions by comparing the actual state of the system with a reference state and then sending a control command to the actuator that is Katsuhiko Ogata. Modem Control Engineering. (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1990) 266.
24 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for proportional to the error -- the difference between the reference state and the actual state. The proportionality constant is referred to as the proportional gain and has units of the signal input to the controller divided by the units of the signal input to the plant. Thus if the signal input to the plant has units of newtons (force) and the feedback and reference signal have units of millimeters, the proportional gain would have units of N/mm. Because the components of the assembly machine are capable of storing energy and there is a very slight delay between when the plant reaches a state and when this state is reported to the controller, the servo system can oscillate. Fortunately, control theory provides a mathematical means to deduce the settling time from a model of the system. The Root-Locus Plot The root-locus plot shows a designer the real and imaginary parts of the roots of the system as function of a variable parameter of the system. Based upon the mathematically modeled transfer function of the plant and the controller, a root-locus plot can be constructed by using an arbitrary velocity gain and by arranging the equations in such a way that the proportional gain becomes the variable parameter. The root-locus will then graphically display the real and imaginary parts of the roots of the system's characteristic equation for all values of the proportional gain between zero and infinity. Since the real component of the root is proportional to the settling time of the system, this technique can be used to estimate the settling time. The proportional gain value that produces the fastest settling time is recorded. The velocity gain is then incremented and the process repeated. By using a search algorithm, the values of the proportional gain and velocity gain that minimize the settling time can be determined..
25 Internally Developed High Performance Assembly Machines Hamlin For second order and lower systems, there is no optimal gain setting. A higher proportional gain setting will always improve the performance if the velocity gain is adjusted to maintain the appropriate damping ratio. For third order and higher systems, there exists a gain setting where a number of the poles possess the same real component. If the locus of one set of poles moves towards the imaginary axis as the proportional gain is increased, and the other set of poles moves away from the imaginary axis, then the point where the two have equal real components is the optimal gain setting. The optimal gain settings for fourth order systems can be determined mathematically. The optimal gain settings for higher order systems require the use of search algorithms. Once the optimal gain settings have been determined, the real component of the dominant poles is recorded. The technique is now repeated for a different set of components. For example, a different motor could be inserted into the model. If the motor's stator is lighter, it will add less mass to the system's structure. If the motor's rotor inertia is lower, it will add less inertia to the moving part of the system. Of course, if it costs more than the original motor, the improved settling time may not be enough to justify this added cost. Although this technique is fairly straight forward, a number of complications arise in its use. The proportional gain provided by the root-locus is usually not the value that is entered into the controller as KI. This occurs because there are a number of other factors that affect the proportional gain of the system. K, therefore must be derived by dividing
26 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for Velocity Feedback X 3 (s) Machine Axis Pulley Figure Block Diagram of X-axis Servo System the optimal gain (from the root locus plot) by the product of the system's other proportional gain terms. The cycle time derived from the real part of the root also assumes that the system is linear. While this assumption holds for many machines in a limited range, a check must be performed to insure that the system remains within this range. The system is said to saturate when it exceeds its linear operation range. Calculating K, and Avoiding Saturation A diagram of the servo-chain for the Acme Mfg. gantry robot is given in M.L The controller is the first component to introduce a proportional gain to the system. Ignoring all other controller gains, this gain multiplies the error by K, which has units of Volts/Length. Since the controller can only produce an output of -10V to + 10V, if the product of K, and the error exceed this value, the controller will only put out 10V. Thus
27 Internally Developed High Performance Assembly Machines Hamlin the onset of saturation is a function of Kp and the magnitude of the error. In order to determine the error size at the onset of saturation, one must first determine Kp. The next proportional gain encountered in the system is the amplifier gain which is set at a constant 3AmpsNolt. Continuing through the servo chain, the next gain encountered is the Motor Torque Conversion K, which converts the input amperage into a torque. For the x-axis motor, K, = 1.34 lb-in/amp. Converting this to MKS units produces.151 N-m/amp. The gear box then multiplies the torque by 5. The pulley converts the torque into a force by dividing the torque by the radius of the pulley which is.02 m. The proportional gain in the controller should therefore be set equal to: Optimal Gain from the Root Locus = Amplifier Gain x Motor Torque Conversion Gain x Gearbox Gain x Pulley Gain The onset of controller saturation can then be determined by dividing 10V by K,. Whenever, the error exceeds this value, the controller will saturate. Saturation is avoided by specifying a trajectory instead of a final destination. The controller is not the only component in the system that can saturate; therefore, all components that can saturate must be checked. The PWM amplifier produces two non-linearities due to its two current overload protection circuits. The first one is set at 15 amps to insure that the motor never sees more than 15 amps. Notice that this corresponds with a -5V to 5V operating region of the controller due to the amplifiers 3A/V gain. The second limit is a root mean square limit which insures that the motor does not see high currents for long periods of time. The onset of this circuit is set as a
28 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for percentage of the first circuit's limiting value. This circuit then reduces the current delivered to the motor using an exponential functional. When this circuit is set to 50% and the first circuit is set to 15 amps, a constant input signal of 10V produces an output of 15 amps that exponentially decays to 7.5 amps. A constant 2.5V input produces a constant 7.5 amp output. When set this way, the linear operating region of the controller and amplifier is reduced to -2.5V to 2.5V, one fourth the intended operating range. In order for the root locus methodology to accurately predict the settling time, linearity must be insured. As such, the onset of saturation must be divided by 4. Limiting the error to values below this number will insure linear operation. Calculating Unit Manufacturing Cost The time required for the machine to complete a move will be approximately equal to 5 times the inverse of the real part of the root, assuming no saturation, regardless of the length of the move. In control theory, the real part of the root is denoted as a. The settling time can therefore be stated mathematically as : ttotw = ts a In case of saturation, this formula will not apply. Typically saturation is avoided by generating a trajectory. This technique strives to limit the maximum error signal to a value that is within the linear range of the system. The cycle time for a move that uses a trajectory can conservatively be estimated by summing the theoretical time required to complete the trajectory plus the aforementioned settling time. ttotal - ts + ttrajectory = 5 + ttrajec.ry The simplest trajectory to specify is a trapezoidal trajectory. The user speclli-e the acceleration, maximum velocity, and final destination. The controller then determnes
29 Internally Developed High Performance Assembly Machines Hamlin the trajectory. For example, if the acceleration is set to 1 m/sec 2, the maximum velocity to 2 m/sec, and we wish the machine to travel 10m, then the controller will produce the trajectory shown in figure Maximum Velocity = 2 m/sec Trapezoidal Trajectory "' Time (sec) Figure Trapezoidal Trajectory Requiring 7 seconds to move 10 meters The controller integrates the velocity trajectory to obtain a reference position function that is a function of time. The controller calculates the error as the difference between the planned position evaluated at time t and the actual position at time t. Basically, the machine makes a series of small moves without stopping between successive points and therefore reduces the size of the error signal. The cycle time of the machine is calculated by summing the time required to complete all of the positioning steps and the operation-at-point steps. The unit manufacturing costs are calculated based upon the anticipated annual uptime of the machine, the annualized cost of the machine, and the cycle time. For example, if the machine costs $300,000, and is expected to have a life of three years, the
30 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for cost can be converted into an annualized number by finding the uniform series equivalent using the firm's cost of capital. The formula for this is: A=P[ (I i(l i)n_ + i)" - I Where A is the uniform series equivalent, P is the present value, n = the number of periods, and i = the interest rate. If the firm's cost of capital is 10%, the resulting annualized cost will be $120,634. Alternatively, straight line depreciation or accelerated depreciation could be used. The machine's theoretical capacity is then determined by dividing the annual uptime by the cycle time. The unit manufacturing cost is then the annualized cost divided by the machine's capacity. A more accurate cost can be determined by dividing the annualized cost by the firm's actual annual production forthe machine in question. Care must be exercised if the theoretical capacity of the machine vastly exceeds the expected annual production. Basing design decisions upon the cost divided by capacity number is risky if the capacity and actual production volume are significantly different.
31 Internally Developed High Performance Assembly Machines Hamlin Chapter 4 - Application of the Methodoloav Modeling the Plant The simplest method for modeling an assembly machine is lumped parameter modeling. Lumped parameter modeling breaks the assembly machine into masses, dampers, springs, and actuating forces. Figure 4.1 shows the ideal case. N Reaction Force Variable ( Fo Position Figure Idealized Assembly Machine, Lumped Parameter Model A variable force controlled by a controller is used to position a mass. The transfer function of this system is a second order equation as discussed in Appendix 1. Actual machines may have some compliance in the transmission. In other words, there is some flexibility between where the force is being generated and where the position is being measured. The lumped parameter model for this condition is displayed in figure 4.2. ariable ( Fo: Transmission I I II I I I I I I -IT- Position Figure Lumped Parameter Model of a Machine with Transmission Compliance
32 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for The machine structure may also be compliant. Because the generation of force requires that an equal and opposite force also be generated, the structure of the machine must transmit this reaction force to the ground. A well designed machine will accomplish this without perceptible movement. A compliant structure is modeled in figure 4.3. A compliant transmission or a compliant structure increases the order of the system to 4. Mass of Structure Variable Controlled Force Force Reaction Force Structural [7T Compliance I IoitionI Position Figure 4.3 -Lumped Parameter Model of a Machine with a Compliant Structure The combination of both a compliant structure and a compliant transmission results in a sixth order system and is shown in figure 4.4. Structural SCompliance Mass of Structure Reaction Force Force Compliance of Transmission M as s to be Positioned P FT mt hi Position Figure Lumped Parameter Model for a Machine with Compliant Structure and Transmission The Acme Mfg. Gantry Robot design is at best a 6th order system. The lack of structural rigidity created primarily by the four long columns adds at least two orders to the system. The transmission is plagued by a number of flexible components and a lack
33 Internally Developed High Performance Assembly Machines Hamlin of symmetry. In the case of the x-axis, the possible sources of compliance in the drive system are the motor shaft, the gear reducer shafts, the gear teeth, the drive shaft, the belt teeth, and the urethane belts. This particular machine is extremely difficult to model because the side nearest to the drive motor and gearbox (the near side) does not encounter the compliance introduced by the torsional twist of the connecting drive shaft that turns the other side (the far side). The net effect is that both sides do not receive the same force input at the same time. The unequal forces cause one carriage to accelerate faster than the other, producing skewing of the y-axis. This skewing motion is exacerbated by the y-axis position of the machine. As the y-axis carriage deviates from the center of the y-axis, one of the x-axis carriages will be carrying a considerably higher percentage of the mass than the other x-axis carriage. As the y-axis is forced out of an orthogonal orientation relative to the x-axis, the horizontal loading on the linear bearings is increased. This increases the viscous, dynamic and static friction acting in the x direction. From a modeling perspective, the assumption is made that this friction is viscous and the damping is constant. Clearly this assumption does not hold for the gantry robot, but the effect may be within an acceptable degree of error. From a controls perspective, it means that the tuning of the controller gains must be robust enough to tolerate the changes in damping. Frequently, the need to maintain controllability over a wide range of damping results in a lowering of the proportional gain and a longer settling time.
34 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for In addition to the variable damping coefficients, the belts cause the spring rates acting on the carriages to vary. This phenomenon occurs because the stiffness of the belt is inversely proportional to its length. Not only are the spring rates variable, but they are also anisotropic. The belt has a finite stiffness when it is loaded in tension but has a very low or zero stiffness when loaded in compression. Looking at the near side carriage, it is apparent upon close inspection that the belt length between the drive pulley and the carriage is always shorter for the piece pulling the carriage towards the motor than for the piece that pulls the carriage away from the motor. As the y-axis traverses the x-axis, one piece of the belt is getting longer -- hence more compliant -- while the other piece is becoming shorter -- hence stiffer. Contrast this behavior with a ballscrew which has equal stiffness in tension and compression. The linear realm of lumped parameter modeling does not provide a means of coping with the variable stiffness or the anisotropy. Clearly, this machine would never appear in a textbook explaining lumped parameter modeling. In order to model this machine, a number of assumptions were necessary to fit a linear model. These assumptions were: 1. No static or dynamic friction 2. Constant damping terms 3. Constant spring rates Although these assumptions were clearly overly optimistic, it was hoped that the machine would exhibit quasi-linear behavior for short moves because the belt length changes would be minimized. Also, by conducting tests at the far end of the machine, the two
35 Internally Developed High Performance Assembly Machines Hamlin pieces of the belt would be as near equal as possible. The lumped parameter model displayed in figure 4.5 was used to construct a transfer function of the machine's x-axis. Lumped Parameter Model V kl bl Structural Compliance ml Xml b2 '- V b4 Viscous Damping in Linear Bearings m2 I-13 Inertia of Motor Rotor and Gearbox Mass to be moved Reflected to Output Structure and Motor EMF and Shaft of Gearbox Motor Stator Viscous Damping of Motor and Gearbox Belt Compliance k2 b4 m3 I Figure Lumped Parameter Model of Gantry Robot X,(s) F(s) -m 2 m 3 s 4 - (b 3 m 2 + b 3 m 3 + b 4 m 2 )s 3 - (m 2 k 2 + m 3 k 2 )s 2 mlm 2 m 3 s 6 + as 5 + a 2 s 4 + a 3 s 3 + a 4 s 2 + (b 2 kk 2 + b 4 kk 2 )s X 3 (s) F(s) (b 3 ml-b 4 m 2 )s 3 + (bjb 3 + k 2,m)s 2 + (b,k 2 + b 3 k,)s + k 1 k 2 mlm 2 m 3 s 6 + as 5 + a 2 s 4 + a 3 s 3 + a 4 s 2 + (b 2 kk 2 + b 4 klk 2 )s Where: a, = b 2 mm 3 + b 3 mm, 2 + b 3 mm, 3 + b 4 M I M 2 + b 1 m 2 m 3 + b 2 m 2 m 3 + b 4 m 2 m 3 a 2 = b 2 b 3 m, + bb 4 m, + b 3 b 4 m, + k 2,mm 2 + k 2 m,m, + bab 2 m, + bib 3 m, +blb 3 m 3 + bb 4 m 2 + b 2 b 3 m, + b 2 b 3 m 3 + b 2 b 4 m 2 + b 2 b 4 m 3 + b 3 b 4 m, +b 3 b 4 m 3 + kmm, a 3 = b 2 k 2 m, + b 4 k 2 m, + bjb 2 b 3 + btb 2 b 4 + blb 3 b4 + b,m, + bjk 2 m, + b 2 k 2 m, + b 2 k 2 m 3 + a 4 = b 4 k 2 m 2 + b 4 k 2 m 3 + b 2 kim 3 + b 3 km, + b 3 km, + b 4 k,m 2 bb 2 k 2 + blb 4 k 2 + b 2 b 3 k, + b 2 b 4 k, + b 3 b 4 k, + klk 2 m 2 + kk2m Figure Transfer Functions for the Structure and Carriage of the Gantry Robot The transfer functions for mass 1 and mass 3 are shown in figure 4.6.
36 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for Identification of the Plant Once the model has been constructed, the next step is to determine the actual values for the design parameters: b,, b 2, b 3, b 4, k 1, k2, m,, m 2, & i 3. The masses are the easiest to determine, and their masses were identified by weighing the components associated with each of the three masses in the model. The stiffness of the structure, k 1, was identified by using a spring scale to act as a force gauge and a dial indicator to measure the deflection. A number of forces were applied to the structure and the deflections were recorded. The forces were then plotted versus the deflections, a line was fit, and the slope of the line was used as k 1. The structural damping, b,, was determined by using the logarithmic decrement method. This method requires the measuring of the structure's response to an impulse. This is accomplished by hitting the structure with a small hammer and then measuring the position of the structure as a function of time. The difference in magnitude of two successive wave forms are used to calculate the damping constant. The belt stiffness was determined by using an Instron tensile tester to determine the spring constant. It is believed that the value determined in this manner overstates the effective value. This discrepancy arises because the drive belt is rigidly clamped on both sides for tensile testing, whereas in use, only the teeth of the belt engage the drive pulley. The other damping constants were estimated from manufacturers' data. Modeling the Controller The compliance in the structure also increased the difficulty of modeling the closed loop feedback. The linear encoder measures the lineal difference between the
37 Internally Developed High Performance Assembly Machines Hamlin structure and the carriage and is therefore feeding back the difference between x 3 and x,. To account for this effect, the block diagram is modified to account for the movement of the base as well as the carriage. In control jargon, the carriage that is moved by the servo system is referred to as the plant. Hence, its transfer function is labeled Gp(s). The transfer function for the base is labeled Gb.e(s). The closed loop transfer function is then solved using block diagram algebra. The resulting closed loop transfer function, G(s), is given at the bottom of figure 4.7. Closed Loop Transfer Function X3(s) G(s) = = Yr(s) Gc(Gp + Gbase) 1 + GcGpH Figure Closed Loop Transfer Function of Gantry Robot Using the Root Locus to Determine Kp, K,, and the Settling Time The resulting transfer function can be converted from the s operator domain to the time domain by taking the inverse LaPlace Transform of the transfer function as was done in Appendix 1. This technique produces a series of sine, cosine, and exponential functions of time that are the system's response to a unit impulse input. This equation
38 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for can be integrated to determine the system's response to a unit step function. The 2% criteria could then be applied to determine the settling time. Fortunately, there is an easier way. The closed loop poles are the roots of the characteristic equation. Since the characteristic equation changes as the proportional gain is varied, the roots of the characteristic equation change. W.R. Evans invented a graphical technique known as the root locus method that shows the location of the poles in the real-imaginary plane as a function of a parameter of the system. Because the real part of the root is proportional to the settling time of the system, this method may be used to estimate settling times. The further to the left of the imaginary axis the pole is located, the lower the settling time. Poles located on the imaginary axis never damp out and poles located to the right of the axis exhibit oscillations whose magnitude increases as a function of time. For fourth order and higher systems, there exists at least one combination of proportional gain and velocity gain that results in the four slowest poles being located the same distance from the y-axis. Because two of the poles are headed in the positive direction and two of the poles are headed in the negative direction, the gain values where both pairs are located the same distance from the imaginary axis are the optimal gain values. If either gain is increased or decreased, one of the two pairs of poles will mor e closer to the imaginary axis and thus yield a less optimal settling time. Figure 4.8 displays the 6 poles and five zeroes for the lumped parameter model when controlled by the controller shown in figure 4.7. In order to verify the model..u
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40 A Strategic Approach to Assessing Needs and Engineering Cost/Performance Tradeoffs for experiment was conducted to compare the lumped parameter model's step response to the step response of the actual robot. The step response of the model is given in figure 4.9 while the actual step response is given in figure Numerous attempts were made to improve the correlation between the model and the actual machine. Although the lumped parameter model accurately predicts the overshoot and does a reasonable job of predicting the next two peaks, the settling time is approximately twice that shown in the actual step response of figure Additionally, the step response given in figure 4.9 was achieved by doubling the known value of mass 2. The number 2 mass was doubled in an attempt to account for some of the non-linear effects displayed by the machine. The rationale behind the doubling is covered in Appendix II. Because the correct weight for mass 2 gave the correct settling time but incorrect overshoot and the doubling of mass 2 yielded the correct overshoot but the incorrect settling time, it was concluded that non-linear effects severely affected the machine's response. Based upon this anecdotal evidence, it was concluded that the dynamic friction and other non-linearities were too severe for the assumptions made in the previous section. Due to the difficulty in estimating the design parameters used in the transfer function, it should be noted that the entire transfer function for the machine can be determined experimentally once the machine has been built. Bode Plots There are a number of techniques for directly deriving the transfer function from experimental data. The simplest technique is to use a Hewlett-Packard digital signal analyzer with the optional transfer function package. This fairly expensive piece of test 40